Commit 53b3b48cfde788f3f9debe40228b3edfc0f823ad

Authored by Francisco Coelho
1 parent 237d62bd
Exists in master

Started pre-paper, to arXiv, maybe?

.gitignore 0 → 100644
... ... @@ -0,0 +1,301 @@
  1 +## Core latex/pdflatex auxiliary files:
  2 +*.aux
  3 +*.lof
  4 +*.log
  5 +*.lot
  6 +*.fls
  7 +*.out
  8 +*.toc
  9 +*.fmt
  10 +*.fot
  11 +*.cb
  12 +*.cb2
  13 +.*.lb
  14 +
  15 +## Intermediate documents:
  16 +*.dvi
  17 +*.xdv
  18 +*-converted-to.*
  19 +# these rules might exclude image files for figures etc.
  20 +# *.ps
  21 +# *.eps
  22 +# *.pdf
  23 +
  24 +## Generated if empty string is given at "Please type another file name for output:"
  25 +.pdf
  26 +
  27 +## Bibliography auxiliary files (bibtex/biblatex/biber):
  28 +*.bbl
  29 +*.bcf
  30 +*.blg
  31 +*-blx.aux
  32 +*-blx.bib
  33 +*.run.xml
  34 +
  35 +## Build tool auxiliary files:
  36 +*.fdb_latexmk
  37 +*.synctex
  38 +*.synctex(busy)
  39 +*.synctex.gz
  40 +*.synctex.gz(busy)
  41 +*.pdfsync
  42 +
  43 +## Build tool directories for auxiliary files
  44 +# latexrun
  45 +latex.out/
  46 +
  47 +## Auxiliary and intermediate files from other packages:
  48 +# algorithms
  49 +*.alg
  50 +*.loa
  51 +
  52 +# achemso
  53 +acs-*.bib
  54 +
  55 +# amsthm
  56 +*.thm
  57 +
  58 +# beamer
  59 +*.nav
  60 +*.pre
  61 +*.snm
  62 +*.vrb
  63 +
  64 +# changes
  65 +*.soc
  66 +
  67 +# comment
  68 +*.cut
  69 +
  70 +# cprotect
  71 +*.cpt
  72 +
  73 +# elsarticle (documentclass of Elsevier journals)
  74 +*.spl
  75 +
  76 +# endnotes
  77 +*.ent
  78 +
  79 +# fixme
  80 +*.lox
  81 +
  82 +# feynmf/feynmp
  83 +*.mf
  84 +*.mp
  85 +*.t[1-9]
  86 +*.t[1-9][0-9]
  87 +*.tfm
  88 +
  89 +#(r)(e)ledmac/(r)(e)ledpar
  90 +*.end
  91 +*.?end
  92 +*.[1-9]
  93 +*.[1-9][0-9]
  94 +*.[1-9][0-9][0-9]
  95 +*.[1-9]R
  96 +*.[1-9][0-9]R
  97 +*.[1-9][0-9][0-9]R
  98 +*.eledsec[1-9]
  99 +*.eledsec[1-9]R
  100 +*.eledsec[1-9][0-9]
  101 +*.eledsec[1-9][0-9]R
  102 +*.eledsec[1-9][0-9][0-9]
  103 +*.eledsec[1-9][0-9][0-9]R
  104 +
  105 +# glossaries
  106 +*.acn
  107 +*.acr
  108 +*.glg
  109 +*.glo
  110 +*.gls
  111 +*.glsdefs
  112 +*.lzo
  113 +*.lzs
  114 +*.slg
  115 +*.slo
  116 +*.sls
  117 +
  118 +# uncomment this for glossaries-extra (will ignore makeindex's style files!)
  119 +# *.ist
  120 +
  121 +# gnuplot
  122 +*.gnuplot
  123 +*.table
  124 +
  125 +# gnuplottex
  126 +*-gnuplottex-*
  127 +
  128 +# gregoriotex
  129 +*.gaux
  130 +*.glog
  131 +*.gtex
  132 +
  133 +# htlatex
  134 +*.4ct
  135 +*.4tc
  136 +*.idv
  137 +*.lg
  138 +*.trc
  139 +*.xref
  140 +
  141 +# hyperref
  142 +*.brf
  143 +
  144 +# knitr
  145 +*-concordance.tex
  146 +# TODO Uncomment the next line if you use knitr and want to ignore its generated tikz files
  147 +# *.tikz
  148 +*-tikzDictionary
  149 +
  150 +# listings
  151 +*.lol
  152 +
  153 +# luatexja-ruby
  154 +*.ltjruby
  155 +
  156 +# makeidx
  157 +*.idx
  158 +*.ilg
  159 +*.ind
  160 +
  161 +# minitoc
  162 +*.maf
  163 +*.mlf
  164 +*.mlt
  165 +*.mtc[0-9]*
  166 +*.slf[0-9]*
  167 +*.slt[0-9]*
  168 +*.stc[0-9]*
  169 +
  170 +# minted
  171 +_minted*
  172 +*.pyg
  173 +
  174 +# morewrites
  175 +*.mw
  176 +
  177 +# newpax
  178 +*.newpax
  179 +
  180 +# nomencl
  181 +*.nlg
  182 +*.nlo
  183 +*.nls
  184 +
  185 +# pax
  186 +*.pax
  187 +
  188 +# pdfpcnotes
  189 +*.pdfpc
  190 +
  191 +# sagetex
  192 +*.sagetex.sage
  193 +*.sagetex.py
  194 +*.sagetex.scmd
  195 +
  196 +# scrwfile
  197 +*.wrt
  198 +
  199 +# svg
  200 +svg-inkscape/
  201 +
  202 +# sympy
  203 +*.sout
  204 +*.sympy
  205 +sympy-plots-for-*.tex/
  206 +
  207 +# pdfcomment
  208 +*.upa
  209 +*.upb
  210 +
  211 +# pythontex
  212 +*.pytxcode
  213 +pythontex-files-*/
  214 +
  215 +# tcolorbox
  216 +*.listing
  217 +
  218 +# thmtools
  219 +*.loe
  220 +
  221 +# TikZ & PGF
  222 +*.dpth
  223 +*.md5
  224 +*.auxlock
  225 +
  226 +# titletoc
  227 +*.ptc
  228 +
  229 +# todonotes
  230 +*.tdo
  231 +
  232 +# vhistory
  233 +*.hst
  234 +*.ver
  235 +
  236 +# easy-todo
  237 +*.lod
  238 +
  239 +# xcolor
  240 +*.xcp
  241 +
  242 +# xmpincl
  243 +*.xmpi
  244 +
  245 +# xindy
  246 +*.xdy
  247 +
  248 +# xypic precompiled matrices and outlines
  249 +*.xyc
  250 +*.xyd
  251 +
  252 +# endfloat
  253 +*.ttt
  254 +*.fff
  255 +
  256 +# Latexian
  257 +TSWLatexianTemp*
  258 +
  259 +## Editors:
  260 +# WinEdt
  261 +*.bak
  262 +*.sav
  263 +
  264 +# Texpad
  265 +.texpadtmp
  266 +
  267 +# LyX
  268 +*.lyx~
  269 +
  270 +# Kile
  271 +*.backup
  272 +
  273 +# gummi
  274 +.*.swp
  275 +
  276 +# KBibTeX
  277 +*~[0-9]*
  278 +
  279 +# TeXnicCenter
  280 +*.tps
  281 +
  282 +# auto folder when using emacs and auctex
  283 +./auto/*
  284 +*.el
  285 +
  286 +# expex forward references with \gathertags
  287 +*-tags.tex
  288 +
  289 +# standalone packages
  290 +*.sta
  291 +
  292 +# Makeindex log files
  293 +*.lpz
  294 +
  295 +# xwatermark package
  296 +*.xwm
  297 +
  298 +# REVTeX puts footnotes in the bibliography by default, unless the nofootinbib
  299 +# option is specified. Footnotes are the stored in a file with suffix Notes.bib.
  300 +# Uncomment the next line to have this generated file ignored.
  301 +#*Notes.bib
... ...
code/asp/bachelor.lp
1 1 person(joey).
2   -%married(X); -married(X) :- person(X).
  2 +married(X); -married(X) :- person(X).
3 3 male(X); female(X) :- person(X).
4 4 bachelor(X) :- male(X), not married(X).
5 5 \ No newline at end of file
... ...
code/asp/color.lp
... ... @@ -9,7 +9,7 @@ edge(5,6). edge(6,5).
9 9  
10 10 col(r). col(b). col(g).
11 11  
12   -1 { color(X,C) : col(C)} 1 :- node(X).
  12 +1 { color(X,C) : col(C) } 1 :- node(X).
13 13 :- edge(X,Y), color(X,C), color(Y,C).
14 14  
15 15 #show color/2.
16 16 \ No newline at end of file
... ...
code/asp/disj.lp 0 → 100644
... ... @@ -0,0 +1,2 @@
  1 +a ; -a.
  2 +b ; c :- a.
0 3 \ No newline at end of file
... ...
code/asp/py_01.lp
... ... @@ -10,11 +10,13 @@ def gcd(a, b):
10 10 else:
11 11 na = a.number
12 12 nb = b.number
13   - nc = nb % na
14   - return gcd(N(nc), a)
  13 + nc = N(nb % na)
  14 + return gcd(nc, a)
15 15  
16 16 #end.
17 17  
18 18 p(210, 213).
19 19 p(1364, 385).
20   -gcd(X, Y, @gcd(X,Y)) :- p(X, Y).
21 20 \ No newline at end of file
  21 +gcd(X, Y, @gcd(X,Y)) :- p(X, Y).
  22 +
  23 +#show gcd/3.
22 24 \ No newline at end of file
... ...
code/asp/tutotial_01.lp
... ... @@ -2,3 +2,4 @@ p(1;2).
2 2 q(1;2).
3 3 r(X,Y) :- p(X), q(Y), X != Y.
4 4  
  5 +#show r/2.
5 6 \ No newline at end of file
... ...
msc/Empty File
task_01/proposal.pdf
No preview for this file type
task_01/proposal.tex 0 → 100644
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  1 +\documentclass[a4paper]{article}
  2 +
  3 +\usepackage{commath}
  4 +\usepackage{hyperref}
  5 +\hypersetup{
  6 + colorlinks=true,
  7 + linkcolor=blue,
  8 +}
  9 +
  10 +\title{Zugzwang\\\textit{Logic and Artificial Intelligence}}
  11 +\author{Francisco Coelho\\ \texttt{fc@uevora.pt}}
  12 +
  13 +\begin{document}
  14 +\maketitle
  15 +
  16 +\begin{abstract}
  17 + A major limitation of logical representations is the implicit assumption that the Background Knowledge (BK) is perfect. This assumption is problematic if data is noisy, which is often the case. Here we aim to explore how ASP specifications with probabilistic facts can lead to characterizations of probability functions on the specification's domain.
  18 +\end{abstract}
  19 +
  20 +\section{Introduction and Motivation }
  21 +
  22 +Answer Set Programming (ASP) is a logic programming paradigm based on the Stable Model semantics of Normal Logic Programs (NP) that can be implemented using the latest advances in SAT solving technology. ASP is a truly declarative language that supports language constructs such as disjunction in the head of a clause, choice rules, and hard and weak constraints.
  23 +
  24 +The Distribution Semantics (DS) is a key approach to extend logical representations with probabilistic reasoning. Probabilistic Facts (PF) are the most basic stochastic DS primitive and they take the form of logical facts, $a$, labelled with a probability, such as $p::a$; Each probabilistic fact represents a boolean random variable that is true with probability $p$ and false with probability $1 - p$. A (consistent) combination of the PFs defines a \textit{total choice} $\theta = \set{p::a, \ldots}$ such that
  25 +
  26 +\begin{equation}
  27 + P(\theta) = \prod_{a\in\theta} p \prod_{a \not\in \theta} (1- p).
  28 +\end{equation}
  29 +
  30 +Our goal is to extend this probability, from total choices, to cover the specification domain. We can foresee two key applications of this extended probability:
  31 +
  32 +\begin{enumerate}
  33 + \item Support any probabilistic reasoning/task on the specification domain.
  34 + \item Also, given a dataset and a divergence measure, now the specification can be scored (by the divergence w.r.t. the \emph{empiric} distribution of the dataset), and sorted amongst other specifications. This is a key ingredient in algorithms searching, for example, an \textit{optimal specification} of the dataset.
  35 +\end{enumerate}
  36 +
  37 +This goal faces a critical problem concerning situations where \textit{multiple} standard models result from a given total choice, illustrated by the following example. The specification
  38 +$$
  39 +\begin{aligned}
  40 + 0.3::a&,\cr
  41 + b \vee c& \leftarrow a.
  42 +\end{aligned}
  43 +$$
  44 +has three stable models, $\set{\neg a}, \set{a, b}$ and $\set{a, c}$. While it is straightforward to set $P(\neg a)=0.7$, there is \textit{no further information} to assign values to $P(a,b)$ and $P(a,c)$. At best, we can use a parameter $\alpha$ such that
  45 +$$
  46 +\begin{aligned}
  47 +P(a,b) &= 0.3 \alpha,\cr
  48 +P(a,c) &= 0.3 (1 - \alpha).
  49 +\end{aligned}
  50 +$$
  51 +
  52 +This uncertainty in inherent to the specification, but can be mitigated with the help of a dataset: the parameter $\alpha$ can be estimated from the empirical distribution.
  53 +
  54 +In summary, if an ASP specification is intended to describe some system that can be observed then:
  55 +
  56 +\begin{enumerate}
  57 + \item The observations can be used to estimate the value of the parameters (such as $\alpha$ above and others entailed from further clauses).
  58 + \item With a probability set for the stable models, we want to extend it to all the samples (\textit{i.e.} consistent sets of literals) of the specification.
  59 + \item This extended probability can then be related to the \textit{empirical distribution}, using a probability divergence, such as Kullback-Leibler; and the divergence value used as a \textit{performance} measure of the specification with respect to the observations.
  60 + \item If that specification is only but one of many possible candidates then that performance measure can be used, \textit{e.g.} as fitness, by algorithms searching (optimal) specifications of a dataset of observations.
  61 +\end{enumerate}
  62 +
  63 +Currently, we are on the step two above: Extending a probability function (with parameters such as $\alpha$), defined on the stable sets of a specification, to all the samples of the specification. This extension must, of course, respect the axioms of probability so that probabilistic reasoning is consistent with the ASP specification.
  64 +
  65 +\section{Work Plan}
  66 +
  67 +A team of two researchers and a undergraduate, master, or Ph.D. student, working over six months with adequate resources, should be able to advance substantial contributions and produce an intermediate progress report for a workshop, a final comprehensive paper for a conference, or start a Ph.D. project with greater reach and depth, describing:
  68 +
  69 +\begin{itemize}
  70 + \item The formalization of the methods outlined above, including the parameter estimation from observations and the probability distribution over the specification samples.
  71 + \item Application and evaluation of this approach, using tools such as \hyperlink{https://ciao-lang.org/playground/scasp.html}{s(casp)}, or the \hyperlink{https://potassco.org/}{Potassco suit} to a range of problems from the simple \textit{Burglar, Earthquake, Alarm} to measuring a specification accuracy on a given dataset, or finding an optimal specification for a given dataset given some background knowledge.
  72 +\end{itemize}
  73 +
  74 +While the theoretical word for this project has yet to be completed, there are some relevant tasks that, with different levels of ambition, can be solved right now: \marginpar{Estas tarefas precisam de maior\ldots\ folego.}
  75 +\begin{enumerate}
  76 + \item \textit{Extract Probability Annotations}. For example, convert the annotated specification \verb!0.3::a. b ; c :- a.! to \verb! a ; -a. b ; c :- a!. This is a simple, syntactical task that can be implemented either with \texttt{prolog} or using \texttt{python} and the API provided by the Potassco suite.
  77 + \item \textit{Extended Probability to Stable Models}. Application of the method outlined before, where the probability of total choices is extended to standard models using parameters, which are next estimated with a dataset.
  78 + \item \textit{Relate Samples, Stable Models and Total Choices}. Determine which stable models, or total choices, contain and which are contained in a given sample.
  79 +\end{enumerate}
  80 +
  81 +\section*{References}
  82 +
  83 +
  84 +\begin{enumerate}
  85 + \item Victor Verreet, Vincent Derkinderen, Pedro Zuidberg Dos Martires, Luc De Raedt, Inference and Learning with Model Uncertainty in Probabilistic Logic Programs (2022)
  86 + \item Andrew Cropper, Sebastijan Dumancic, Richard Evans, Stephen H. Muggleton, Inductive logic programming at 30 (2021)
  87 + \item Fabio Gagliardi Cozman, Denis Deratani Mauá, The joy of Probabilistic Answer Set Programming: Semantics - complexity, expressivity, inference (2020)
  88 + \item Fabrizio Riguzzi, Foundations of Probabilistic Logic Programming Languages, Semantics, Inference and Learning. Rivers Publishers (2018)
  89 + \item Martin Gebser, Roland Kaminski, Benjamin Kaufmann, and Torsten Schaub, Answer Set Solving in Practice, Morgan \& Claypool Publishers (2013)
  90 +\end{enumerate}
  91 +
  92 +\end{document}
0 93 \ No newline at end of file
... ...
text/00_PASP.aux
... ... @@ -1,170 +0,0 @@
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text/00_PASP.pdf
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text/00_PASP.snm
... ... @@ -1,14 +0,0 @@
1   -\beamer@slide {eq:prob.tc}{5}
2   -\beamer@slide {prop:unique.ext.tcsm}{7}
3   -\beamer@slide {prop:lucases}{13}
4   -\beamer@slide {prop:lucases.a}{13}
5   -\beamer@slide {prop:lucases.b}{13}
6   -\beamer@slide {prop:lucases.c}{13}
7   -\beamer@slide {prop:lucases.d}{13}
8   -\beamer@slide {def:w.inconsistent}{16}
9   -\beamer@slide {eq:prob.sm}{16}
10   -\beamer@slide {def:w.disj}{16}
11   -\beamer@slide {def:w.conj}{16}
12   -\beamer@slide {def:w.empty}{16}
13   -\beamer@slide {eq:def.prob}{17}
14   -\beamer@slide {eq:def.prob.event}{17}
text/00_PASP.synctex.gz
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text/00_PASP.tex
... ... @@ -23,27 +23,48 @@
23 23 \usepackage[overridenote]{pdfpc}
24 24  
25 25 \usepackage{tikz}
  26 +\usetikzlibrary{
  27 + % shapes,
  28 + % arrows,
  29 + % backgrounds,
  30 + positioning,
  31 +}
26 32 \tikzset{
27   -rrect/.style={
28   - rectangle,
29   - rounded corners,
30   - fill=gray!10,
31   - % minimum width=1em,
32   - % text width=2em,
33   - % draw=blue!75,
34   - % very thick,
35   - align=center,
36   - }
  33 + sm/.style={
  34 + rectangle,
  35 + rounded corners,
  36 + fill=teal!10,
  37 + align=center,
  38 + minimum height=1.5em,
  39 + },
  40 + sup/.style={
  41 + rectangle,
  42 + rounded corners,
  43 + fill=blue!10,
  44 + align=center,
  45 + minimum height=1.5em,
  46 + },
  47 + sub/.style={
  48 + rectangle,
  49 + rounded corners,
  50 + fill=red!10,
  51 + align=center,
  52 + minimum height=1.5em,
  53 + },
  54 + ind/.style={
  55 + align=center,
  56 + minimum height=1.5em,
  57 + },
37 58 }
38 59  
39   -\usepackage{commath}
40   -\usepackage{amssymb}
41 60 \usepackage[T1]{fontenc}
42 61 \usepackage{hyperref}
43 62 \hypersetup{%
44 63 colorlinks=true,
45 64 allcolors=blue,
46 65 }
  66 +\usepackage{commath}
  67 +\usepackage{amssymb}
47 68 %
48 69 % Local commands
49 70 %
... ... @@ -89,11 +110,11 @@ rrect/.style={
89 110 % --------------------------------
90 111 \item $\co{x} = 1 - x$.
91 112 % --------------------------------
92   - \item \textbf{Probabilistic Atomic Choice (PAC):} $\alpha :: a$ defines $a \lor \neg a$ and probabilities $\pr{a} = \alpha, \pr{\neg a} = \co{\alpha}$.
  113 + \item \textbf{Probabilistic Atomic Choice (PAC):} $x :: a$ defines $a \lor \neg a$ and probabilities $\pr{a} = x, \pr{\neg a} = \co{x}$.
93 114 % --------------------------------
94   - \item $\delta a$ denotes $a \lor \neg a$ and $\delta\! \set{\alpha :: a, a \in \fml{A}} = \set{\delta a, a \in \fml{A}}$ for a set of atoms $\fml{A}$.
  115 + \item $\delta a$ denotes $a \lor \neg a$ and $\delta\! \set{x :: a, a \in \fml{A}} = \set{\delta a, a \in \fml{A}}$ for a set of atoms $\fml{A}$.
95 116 % --------------------------------
96   - \item \textbf{Closed World Assumption:} $\naf x \models \neg x$.
  117 + \item \textbf{Closed World Assumption:} $\naf p \models \neg p$.
97 118 % --------------------------------
98 119 % \item Probabilistic choices and sub-goals are independent.
99 120 % --------------------------------
... ... @@ -106,25 +127,24 @@ rrect/.style={
106 127 \begin{itemize}
107 128 % --------------------------------
108 129 \item \textbf{Atoms} $\fml{A}$,
109   - $\overline{\fml{A}} = \cset{\neg a}{a \in \fml{A}}$,
  130 + $\overline{\fml{A}} = \cset{\neg a}{a \in \fml{A}}$, and \textbf{literals} $\fml{L} = \fml{A} \cup \co{\fml{A}}$.
110 131 % --------------------------------
111   - \item \textbf{Samples} $\fml{Z}$:
112   - $$\fml{Z} = \cset{z = \alpha \cup \beta }{ \alpha \subseteq \fml{A} \land \beta \subseteq \overline{\fml{A}} }$$
  132 + \item \textbf{Samples} $z \in \fml{Z} \iff z \subseteq \fml{L}$.
113 133 % --------------------------------
114   - \item \textbf{Interpretations} or \textit{consistent samples} $\fml{I}$ :
115   - $$\fml{I} = \cset{z \in \fml{Z} }{ \forall a \in \fml{A}~\envert{\set{a,\neg a} \cap z} \leq 1}.$$
  134 + \item \textbf{Events} or \textit{consistent samples} $\fml{E}$ :
  135 + $$\fml{E} = \cset{z \in \fml{Z} }{ \forall a \in \fml{A}~\envert{\set{a,\neg a} \cap z} \leq 1}.$$
116 136 % --------------------------------
117 137 \item \textit{PASP Problem} or \textbf{Specification:} $P = C \land F \land R$ where
118 138 % --------------------------------
119 139 \begin{itemize}
120 140 % --------------------------------
121   - \item $C = C_P = \cset{\alpha_i :: a_i }{ i \in 1:n \land a_i \in \fml{A}}$ \textit{pacs}.
  141 + \item $C = C_P = \cset{x_i :: a_i }{ i \in 1:n \land a_i \in \fml{A}}$ \textit{pacs}.
122 142 % --------------------------------
123 143 \item $F = F_P$ \textit{facts}.
124 144 % --------------------------------
125 145 \item $R = R_P$ \textit{rules}.
126 146 % --------------------------------
127   - \item $\fml{A}_P, \fml{Z}_P$ and $\fml{I}_P$: \textit{atoms}, \textit{samples} and \textit{interpretations} of $P$.
  147 + \item $\fml{A}_P, \fml{Z}_P$ and $\fml{E}_P$: \textit{atoms}, \textit{samples} and \textit{events} of $P$.
128 148 \end{itemize}
129 149 % --------------------------------
130 150 \item \textbf{Stable Models} of $P$, $\fml{S} = \fml{S}_P$, are the stable models of $\delta P = \delta C + F + R$.
... ... @@ -137,7 +157,7 @@ rrect/.style={
137 157 % --------------------------------
138 158 \begin{itemize}
139 159 % --------------------------------
140   - \item \textbf{Total Choices:} $\Theta = \Theta_C = \Theta_P$ elements are $\theta = \cset{t_c}{c \in C}$ where $c=\alpha::a$ and $t_c$ is $a$ or $\neg a$.
  160 + \item \textbf{Total Choices:} $\Theta = \Theta_C = \Theta_P$ elements are $\theta = \cset{t_c}{c \in C}$ where $c=x::a$ and $t_c$ is $a$ or $\neg a$.
141 161 % --------------------------------
142 162 %\item For $s\in\fml{S}$ let $\theta_s \subseteq s$ (unique \textit{total choice})
143 163 %\item Define $\fml{S}_\theta = \cset{s \in \fml{S}}{\theta \subset s}$.
... ... @@ -146,7 +166,7 @@ rrect/.style={
146 166 % --------------------------------
147 167 \item \textbf{Total Choice Probability:}
148 168 \begin{equation}
149   - \pr{\theta} = \prod_{a \in \theta}\alpha \prod_{\neg a \in \theta}\co{\alpha}.\label{eq:prob.tc}
  169 + \pr{\theta} = \prod_{a \in \theta}x \prod_{\neg a \in \theta}\co{x}.\label{eq:prob.tc}
150 170 \end{equation}
151 171 % --------------------------------
152 172 \end{itemize}
... ... @@ -157,7 +177,7 @@ rrect/.style={
157 177 \begin{frame}
158 178 % --------------------------------
159 179 \begin{block}{Problem Statement}
160   - How to \textit{extend} probability from total choices to stable models, interpretations and samples?
  180 + How to \textit{extend} probability from total choices to stable models, events and samples?
161 181 \end{block}
162 182 % --------------------------------
163 183 \begin{quotation}
... ... @@ -193,11 +213,11 @@ rrect/.style={
193 213 % --------------------------------
194 214 \item $\pr{\set{\neg a}} = 0.7$ is straightforward.
195 215 % --------------------------------
196   - \item But, no \textit{informed} choice for $\alpha\in\intcc{0,1}$ in
  216 + \item But, no \textit{informed} choice for $x\in\intcc{0,1}$ in
197 217 $$
198 218 \begin{aligned}
199   - \pr{\set{a, b}} &= 0.3 \alpha, \cr
200   - \pr{\set{a, c}} &= 0.3 \co{\alpha}.
  219 + \pr{\set{a, b}} &= 0.3 x, \cr
  220 + \pr{\set{a, c}} &= 0.3 \co{x}.
201 221 \end{aligned}
202 222 $$
203 223 % --------------------------------
... ... @@ -209,28 +229,43 @@ rrect/.style={
209 229 % --------------------------------
210 230 \begin{itemize}
211 231 % --------------------------------
212   - \item The specification \textit{lacks information} to set $\alpha\in\intcc{0,1}$ in
  232 + \item The specification \textit{lacks information} to set $x\in\intcc{0,1}$ in
213 233 $$
214 234 \begin{aligned}
215   - \pr{\set{a, b}} &= 0.3 \alpha, \cr
216   - \pr{\set{a, c}} &= 0.3 \co{\alpha}.
  235 + \pr{\set{a, b}} &= 0.3 x, \cr
  236 + \pr{\set{a, c}} &= 0.3 \co{x}.
217 237 \end{aligned}
218 238 $$
219   - \item A \textit{random variable} captures this uncertainty:
  239 + \item A \textit{random variable} captures this uncertainty, \alert{assuming} that the stable models are statistically independent:
220 240 $$
221 241 \begin{aligned}
222   - \pr{\set{\neg a} \given A = \alpha } &= 0.7, \cr
223   - \pr{\set{a, b} \given A = \alpha } &= 0.3 \alpha, \cr
224   - \pr{\set{a, c} \given A = \alpha } &= 0.3 \co{\alpha}.
  242 + \pr{\set{\neg a} \given X = x } &= 0.7, \cr
  243 + \pr{\set{a, b} \given X = x } &= 0.3 x, \cr
  244 + \pr{\set{a, c} \given X = x } &= 0.3 \co{x}.
225 245 \end{aligned}
226 246 $$
227   - \item Other uncertainties lead to further parameters:
  247 + \item Other uncertainties may lead to further conditions:
228 248 $$
229   - \pr{s \given A_1 = \alpha_1, \ldots, A_n = \alpha_n }.
  249 + \pr{s \given X_1 = x_1, \ldots, X_n = x_n }.
230 250 $$
231 251 % --------------------------------
232 252 \end{itemize}
233   - Reducing \textbf{specification uncertainty}, \textit{e.g.} setting $A = 0.21$, must result from \textbf{observations}.
  253 + Reducing \textbf{uncertainty}, \textit{e.g.} setting $X = 0.21$, must result from \textbf{external} sources, since the specification lacks information for further assertions.
  254 + % --------------------------------
  255 +\end{frame}
  256 +% ================================================================
  257 +\begin{frame}{Independence of Stable Models}
  258 + % --------------------------------
  259 +
  260 + \begin{itemize}
  261 + \item[Q:] Why are the stable models assumed statistically independent?
  262 + % --------------------------------
  263 + \item[A:] Because dependence can be \textit{explicitly} modelled.
  264 + % --------------------------------
  265 + \item So, it is assumed \textit{intention} of the \textit{modeller} to not explicit express such dependences.
  266 + % --------------------------------
  267 + \item \textbf{For example:} \todo{Some key examples}.
  268 + \end{itemize}
234 269 % --------------------------------
235 270 \end{frame}
236 271 % ================================================================
... ... @@ -239,9 +274,9 @@ rrect/.style={
239 274 A \textit{random variable} captures this uncertainty:
240 275 $$
241 276 \begin{aligned}
242   - \pr{\set{\neg a} \given A = \alpha } &= 0.7, \cr
243   - \pr{\set{a, b} \given A = \alpha } &= 0.3 \alpha, \cr
244   - \pr{\set{a, c} \given A = \alpha } &= 0.3 \co{\alpha}.
  277 + \pr{\set{\neg a} \given X = x } &= 0.7, \cr
  278 + \pr{\set{a, b} \given X = x } &= 0.3 x, \cr
  279 + \pr{\set{a, c} \given X = x } &= 0.3 \co{x}.
245 280 \end{aligned}
246 281 $$
247 282 % --------------------------------
... ... @@ -265,13 +300,13 @@ rrect/.style={
265 300 % --------------------------------
266 301 \end{frame}
267 302 % ================================================================
268   -\begin{frame}%{Specification, Data \& Evaluation}
  303 +\begin{frame}{Leap into Inductive Programming}
269 304 % --------------------------------
270 305 Given a method that produces a distribution of samples, $p$, from a specification, $P$ and:
271 306 % --------------------------------
272 307 \begin{itemize}
273 308 % --------------------------------
274   - \item $Z$, a dataset of samples.
  309 + \item $Z$, a dataset (of samples).
275 310 % --------------------------------
276 311 \item $e$, the respective empirical distribution.
277 312 % --------------------------------
... ... @@ -279,8 +314,18 @@ rrect/.style={
279 314 % --------------------------------
280 315 \end{itemize}
281 316 % --------------------------------
282   - \begin{block}{Motivation}
283   - Then $D\at{P} = D\at{e, p}$ is a \textbf{performance} measure of $P$ and can be used, \textit{e.g.} as fitness, by algorithms searching for \textbf{optimal specifications of a dataset}.
  317 + \begin{block}{Specification Performance \& Inductive Programming}
  318 + % --------------------------------
  319 + \begin{itemize}
  320 + % --------------------------------
  321 + \item $D\at{P} = D\at{e, p}$ is a \textbf{performance} measure of $P$.
  322 + % --------------------------------
  323 + \item Predictor performance measures, such as accuracy, are common in \textit{Machine Learning} tasks.
  324 + % --------------------------------
  325 + \item For \textit{Inductive Programming} this performance can be used, \textit{e.g.} as fitness, by algorithms searching for \textbf{optimal specifications of a dataset}.
  326 + % --------------------------------
  327 + \end{itemize}
  328 + % --------------------------------
284 329 \end{block}
285 330 % --------------------------------
286 331 \end{frame}
... ... @@ -290,18 +335,21 @@ rrect/.style={
290 335 \begin{frame}{Resolution Path}
291 336 Prior to \textit{conciliation} with data:
292 337 \begin{enumerate}
293   - \item Use \textit{conditional parameters} to represent lack of knowledge \emph{e.g.} $\alpha$ in the disjunction example.
294   - \item This extends probability from total choices to \textit{standard models} --- \alert{hopefully}.
295   - \item Assume probability set on standard models; Extend it to \textit{interpretations}. \textbf{How?} Later.
296   - \item Set $\pr{z} = 0$ for $z \in \fml{Z} \setminus \fml{I}$ (inconsistent samples).
  338 + \item \alert{Hopefully}, \textit{conditional parameters} extend probability from total choices to \textit{standard models}.
  339 + \item \textbf{How} to extend it to \textit{events}?
  340 + \begin{itemize}
  341 + \item $\pr{x} = 0$ for $x$ \textit{excluded} by the specification, including \textit{inconsistent} samples.
  342 + \item $\pr{x}$ depends on the $s \in \fml{S}$ that contain/are contained in $x$.
  343 + \end{itemize}
297 344 \end{enumerate}
298   - Assuming a conditional probability set on stable models, \textit{How to extend it to interpretations?}
  345 + \alert{Consider probabilities \textbf{conditional} on the total choice!}
299 346 \end{frame}
300   -\begin{frame}{Bounds of Interpretations}
  347 +% ================================================================
  348 +\begin{frame}{Bounds of Events}
301 349 % --------------------------------
302 350 \begin{itemize}
303 351 % --------------------------------
304   - \item For $x\in\fml{I}$:
  352 + \item For $x\in\fml{E}$:
305 353 % --------------------------------
306 354 \begin{itemize}
307 355 % --------------------------------
... ... @@ -315,13 +363,25 @@ rrect/.style={
315 363 % --------------------------------
316 364 \begin{enumerate}
317 365 % --------------------------------
318   - \item\label{prop:lucases.a} $\lset{x} = \set{x} = \uset{x}$ and $x$ is a stable model.
  366 + \item\label{prop:lucases.a} $\lset{x} = \set{x} = \uset{x}$ and $x$ is a stable model. Then:
  367 + \begin{equation}
  368 + \pr{x \given C = \theta_x} = d\at{x}.
  369 + \end{equation}
319 370 % --------------------------------
320   - \item\label{prop:lucases.b} $\lset{x} \neq \emptyset \land \uset{x} = \emptyset$.
  371 + \item\label{prop:lucases.b} $\lset{x} \neq \emptyset \land \uset{x} = \emptyset$. Then:
  372 + \begin{equation}
  373 + \pr{x \given C = \theta_s, s \in \lset{x}} = \prod_{s\in\lset{x}} d\at{s}.
  374 + \end{equation}
321 375 % --------------------------------
322   - \item\label{prop:lucases.c} $\lset{x} = \emptyset \land \uset{x} \neq \emptyset$.
  376 + \item\label{prop:lucases.c} $\lset{x} = \emptyset \land \uset{x} \neq \emptyset$. Then:
  377 + \begin{equation}
  378 + \pr{x \given C = \theta_s, s \in \uset{x}} = \sum_{s\in\uset{x}} d\at{s}.
  379 + \end{equation}
323 380 % --------------------------------
324   - \item\label{prop:lucases.d} $\lset{x} = \emptyset = \uset{x}$.
  381 + \item\label{prop:lucases.d} $\lset{x} = \emptyset = \uset{x}$. Then:
  382 + \begin{equation}
  383 + \pr{x} = 0.
  384 + \end{equation}
325 385 % --------------------------------
326 386 \end{enumerate}
327 387 because stable models are \textit{minimal}.
... ... @@ -330,57 +390,326 @@ rrect/.style={
330 390 % --------------------------------
331 391 \end{frame}
332 392 % ================================================================
333   -\begin{frame}{Disjunction Example | The Interpretation's Lattice}
  393 +\begin{frame}{Conditional on Total Choices}
  394 + % --------------------------------
  395 + \begin{itemize}
  396 + % --------------------------------
  397 + \item A stable model is entailed by an atomic choice plus the facts and rules of the specification.
  398 + \item We express that entailment as a \textit{conditional}. For example:
  399 + $$\pr{\set{a,b} \given X = x} = \pr{b \given X = x, \Theta = a}\pr{\theta = a}$$
  400 + \item And now $\pr{b \given X = x, \Theta = a} = x$, since $X$ is a proxy for the stable models of the total choice $\theta = a$, we can further.
  401 + % --------------------------------
  402 + \end{itemize}
  403 + % --------------------------------
  404 +\end{frame}
  405 +% ================================================================
  406 +\begin{frame}{Disjunction Example | The Events Lattice}
334 407 \begin{center}
335 408 \begin{tikzpicture}
  409 + % --------------------------------
  410 + %\draw [help lines, color=gray!20] grid (11,7);
  411 + % --------------------------------
  412 + \node at (7, 7) {$\pr{\Theta=a} = 0.3$};
  413 + \node at (7, 6) {$x = \pr{S = ab \given \Theta}$};
  414 + \node at (7, 5) {$\co{x} = \pr{S \not= ab \given \Theta}$};
  415 + \node at (7, 7.5) {$\pr{E = abc \given \Theta} = \pr{S = ab, S = ac \given \Theta }$};
  416 + % --------------------------------
  417 + % \node [rrect] (sub) at (2, 7) {sub};
  418 + % --------------------------------
  419 + % \node [ fill=gray!10] (sup) at (3, 7) {sup};
  420 + % --------------------------------
  421 + % \node (ind) at (4, 7) {ind};
  422 + % --------------------------------
336 423 %
337   - \draw [help lines, color=gray!20] grid (11,7);
338   - %
339   - \node [fill=gray!50] (sm) at (1, 7) {sm};
340   - \node [draw,rrect] (sub) at (2, 7) {sub};
341   - \node [draw, fill=gray!10] (sup) at (3, 7) {sup};
342   - \node (ind) at (4, 7) {ind};
343   - %
344   - \node [draw, rrect] (E) at (5.5,0) {$\emptyset$};
  424 + % --------------------------------
  425 + \node [ sub,
  426 + pin=45:\textcolor{violet}{$1$} ]
  427 + (E) at (5.5,0) {$\emptyset$};
  428 + % --------------------------------
345 429 %
346   - \node [draw, rrect] (a) at (1.5,1.5) {$a$};
347   - \node [draw, rrect] (b) at (0,1.5) {$b$};
348   - \node [draw, rrect] (c) at (4.5,1.5) {$c$};
349   - \node [fill=gray!50] (A) at (8.5,1.5) {$\co{a}$};
350   - \node (B) at (9.5,1.5) {$\co{b}$};
351   - \node (C) at (10.5,1.5) {$\co{c}$};
  430 + % --------------------------------
  431 + \node [ sub,
  432 + pin=0:\textcolor{blue!75}{$1$} ]
  433 + (a) at (1.5,1.5) {$a$};
  434 + % --------------------------------
  435 + \node [ sub,
  436 + pin=315:\textcolor{blue!50}{$x$}]
  437 + (b) at (0,1.5) {$b$};
  438 + % --------------------------------
  439 + \node [ sub,
  440 + pin=315:\textcolor{blue!50}{$\co{x}$}]
  441 + (c) at (4.5,1.5) {$c$};
  442 + % --------------------------------
  443 + % \node [ sm,
  444 + % pin=270:\textcolor{teal}{$1$}]
  445 + % (A) at (8.5,1.5) {$\co{a}$};
  446 + % % --------------------------------
  447 + % \node [ ind,
  448 + % pin=270:\textcolor{purple}{$0$}]
  449 + % (B) at (9.5, 1.5) {$\co{b}$};
  450 + % % --------------------------------
  451 + % \node [ ind,
  452 + % pin=270:\textcolor{purple}{$0$}]
  453 + % (C) at (10.5, 1.5) {$\co{c}$};
  454 + % --------------------------------
352 455 %
353   - \node [fill=gray!50] (ab) at (0,4) {$ab$};
354   - \node [fill=gray!50] (ac) at (3,4) {$ac$};
355   - \node (aB) at (1,4) {$a\co{b}$};
356   - \node (aC) at (2,4) {$a\co{c}$};
357   - \node [draw, fill=gray!10] (Ab) at (4,4) {$\co{a}b$};
358   - \node [draw, fill=gray!10] (Ac) at (5,4) {$\co{a}c$};
359   - \node [draw, fill=gray!10] (AB) at (6,4) {$\co{a}\co{b}$};
360   - \node [draw, fill=gray!10] (AC) at (7,4) {$\co{a}\co{c}$};
361   - \node [fill=white] (bc) at (10,4) {$bc$};
362   - \node [fill=white] (bC) at (11,4) {$b\co{c}$};
363   - \node [fill=white] (Bc) at (12,4) {$\co{b}c$};
364   - \node [fill=white] (BC) at (13,4) {$\co{b}\co{c}$};
  456 + % --------------------------------
  457 + \node [ sm,
  458 + pin=0:\textcolor{teal}{$x$}]
  459 + (ab) at (0,4) {$ab$};
  460 + % --------------------------------
  461 + \node [ sm,
  462 + pin=0:\textcolor{teal}{$\co{x}$}]
  463 + (ac) at (3,4) {$ac$};
  464 + % --------------------------------
  465 + % \node [ ind,
  466 + % pin=270:\textcolor{purple}{$0$}]
  467 + % (aB) at (1,4) {$a\co{b}$};
  468 + % % --------------------------------
  469 + % \node [ ind,
  470 + % pin=270:\textcolor{purple}{$0$}]
  471 + % (aC) at (2,4) {$a\co{c}$};
  472 + % --------------------------------
  473 + % \node [ sup,
  474 + % pin=90:\textcolor{blue!50}{$1$}]
  475 + % (Ab) at (4,4) {$\co{a}b$};
  476 + % % --------------------------------
  477 + % \node [ sup,
  478 + % pin=90:\textcolor{blue!50}{$1$}]
  479 + % (Ac) at (5,4) {$\co{a}c$};
  480 + % % --------------------------------
  481 + % \node [ sup,
  482 + % pin=90:\textcolor{blue!50}{$1$}]
  483 + % (AB) at (6,4) {$\co{a}\co{b}$};
  484 + % % --------------------------------
  485 + % \node [ sup,
  486 + % pin=90:\textcolor{blue!50}{$1$}]
  487 + % (AC) at (7,4) {$\co{a}\co{c}$};
  488 + % % --------------------------------
  489 + % \node [ ind,
  490 + % pin=270:\textcolor{purple}{$0$}]
  491 + % (bc) at (10,4) {$bc$};
  492 + % % --------------------------------
  493 + % \node [ ind,
  494 + % pin=270:\textcolor{purple}{$0$}]
  495 + % (bC) at (11,4) {$b\co{c}$};
  496 + % % --------------------------------
  497 + % \node [ ind,
  498 + % pin=270:\textcolor{purple}{$0$}]
  499 + % (Bc) at (9.5,3.5) {$\co{b}c$};
  500 + % % --------------------------------
  501 + % \node [ ind,
  502 + % pin=270:\textcolor{purple}{$0$}]
  503 + % (BC) at (10.5,3.5) {$\co{b}\co{c}$};
  504 + % --------------------------------
365 505 %
366   - \node [draw, fill=gray!10] (abc) at (1.5,6) {$abc$};
367   - \node (abC) at (3,6) {$ab\co{c}$};
368   - \node (aBc) at (4,6) {$a\co{b}c$};
369   - \node (aBC) at (5,6) {$a\co{b}\co{c}$};
370   - \node [draw, fill=gray!10] (Abc) at (7,6) {$\co{a}bc$};
371   - \node [draw, fill=gray!10] (AbC) at (8,6) {$\co{a}b\co{c}$};
372   - \node [draw, fill=gray!10] (ABc) at (9,6) {$\co{a}\co{b}c$};
373   - \node [draw, fill=gray!10] (ABC) at (10,6) {$\co{a}\co{b}\co{c}$};
  506 + % --------------------------------
  507 + \node [ sup,
  508 + pin=45:\textcolor{blue!75}{$x\co{x}$}]
  509 + (abc) at (1.5,6)
  510 + {$abc$};
  511 + % --------------------------------
  512 + \node [ sup,
  513 + pin=45:\textcolor{blue!50}{$x$}]
  514 + (abC) at (0,6) {$ab\co{c}$};
  515 + % --------------------------------
  516 + \node [ sup,
  517 + pin=45:\textcolor{blue!50}{$\co{x}$}]
  518 + (aBc) at (3,6) {$a\co{b}c$};
  519 + % --------------------------------
  520 + % \node [ ind,
  521 + % pin=90:\textcolor{purple}{$0$}]
  522 + % (aBC) at (5,6) {$a\co{b}\co{c}$};
  523 + % --------------------------------
  524 + % \node [ sup,
  525 + % pin=90:\textcolor{blue!50}{$1$}]
  526 + % (Abc) at (7,6) {$\co{a}bc$};
  527 + % % --------------------------------
  528 + % \node [ sup,
  529 + % pin=270:\textcolor{blue!50}{$1$}]
  530 + % (AbC) at (8,6) {$\co{a}b\co{c}$};
  531 + % % --------------------------------
  532 + % \node [ sup,
  533 + % pin=270:\textcolor{blue!50}{$1$}]
  534 + % (ABc) at (9,6) {$\co{a}\co{b}c$};
  535 + % % --------------------------------
  536 + % \node [ sup,
  537 + % pin=270:\textcolor{blue!50}{$1$}]
  538 + % (ABC) at (10,6) {$\co{a}\co{b}\co{c}$};
  539 + % --------------------------------
374 540 %
  541 + % --------------------------------
375 542 \draw [->] (ab) to [out=270,in=180] (E);
376 543 \draw [->] (ab) to [out=270,in=90] (a);
377 544 \draw [->] (ab) to [out=270,in=90] (b);
378 545 \draw [->] (ab) to [out=90,in=270] (abc);
  546 + \draw [->] (ab) to [out=90,in=270] (abC);
379 547 %
380 548 \draw [->] (ac) to [out=270,in=180] (E);
381 549 \draw [->] (ac) to [out=270,in=90] (a);
382 550 \draw [->] (ac) to [out=270,in=90] (c);
383 551 \draw [->] (ac) to [out=90,in=270] (abc);
  552 + \draw [->] (ac) to [out=90,in=270] (aBc);
  553 + %
  554 + % \draw [->] (A) to [out=270,in=0] (E);
  555 + % %
  556 + % \draw [->] (A) to [out=90,in=270] (Abc);
  557 + % \draw [->] (A) to [out=90,in=270] (AbC);
  558 + % \draw [->] (A) to [out=90,in=270] (ABc);
  559 + % \draw [->] (A) to [out=90,in=270] (ABC);
  560 + % %
  561 + % \draw [->] (A) to [out=90,in=270] (Ab);
  562 + % \draw [->] (A) to [out=90,in=270] (Ac);
  563 + % \draw [->] (A) to [out=90,in=270] (AB);
  564 + % \draw [->] (A) to [out=90,in=270] (AC);
  565 + \end{tikzpicture}
  566 + \end{center}
  567 +\end{frame}
  568 +% ================================================================
  569 +\begin{frame}{Disjunction Example | The Events Lattice}
  570 + \begin{center}
  571 + \begin{tikzpicture}
  572 + % --------------------------------
  573 + %\draw [help lines, color=gray!20] grid (11,7);
  574 + % --------------------------------
  575 + \node [sm] (sm) at (5, 7) {$\pr{\Theta=\set{\co{a}}} = \co{0.3}$};
  576 + % --------------------------------
  577 + % \node [rrect] (sub) at (2, 7) {sub};
  578 + % --------------------------------
  579 + % \node [ fill=gray!10] (sup) at (3, 7) {sup};
  580 + % --------------------------------
  581 + % \node (ind) at (4, 7) {ind};
  582 + % --------------------------------
  583 + %
  584 + % --------------------------------
  585 + \node [ sub,
  586 + pin=45:\textcolor{violet}{$1$} ]
  587 + (E) at (5.5,0) {$\emptyset$};
  588 + % --------------------------------
  589 + %
  590 + % --------------------------------
  591 + % \node [ sub,
  592 + % pin=270:\textcolor{blue!75}{$1$} ]
  593 + % (a) at (1.5,1.5) {$a$};
  594 + % % --------------------------------
  595 + % \node [ sub,
  596 + % pin=270:\textcolor{blue!50}{$x$}]
  597 + % (b) at (0,1.5) {$b$};
  598 + % % --------------------------------
  599 + % \node [ sub,
  600 + % pin=270:\textcolor{blue!50}{$\co{x}$}]
  601 + (c) at (4.5,1.5) {$c$};
  602 + % --------------------------------
  603 + \node [ sm,
  604 + pin=45:\textcolor{teal}{$1$}]
  605 + (A) at (8.5,1.5) {$\co{a}$};
  606 + % --------------------------------
  607 + % \node [ ind,
  608 + % pin=270:\textcolor{purple}{$0$}]
  609 + % (B) at (9.5, 1.5) {$\co{b}$};
  610 + % % --------------------------------
  611 + % \node [ ind,
  612 + % pin=270:\textcolor{purple}{$0$}]
  613 + % (C) at (10.5, 1.5) {$\co{c}$};
  614 + % --------------------------------
  615 + %
  616 + % --------------------------------
  617 + % \node [ sm,
  618 + % pin=90:\textcolor{teal}{$x$}]
  619 + % (ab) at (0,4) {$ab$};
  620 + % % --------------------------------
  621 + % \node [ sm,
  622 + % pin=90:\textcolor{teal}{$\co{x}$}]
  623 + % (ac) at (3,4) {$ac$};
  624 + % % --------------------------------
  625 + % \node [ ind,
  626 + % pin=270:\textcolor{purple}{$0$}]
  627 + % (aB) at (1,4) {$a\co{b}$};
  628 + % % --------------------------------
  629 + % \node [ ind,
  630 + % pin=270:\textcolor{purple}{$0$}]
  631 + % (aC) at (2,4) {$a\co{c}$};
  632 + % --------------------------------
  633 + \node [ sup,
  634 + pin=135:\textcolor{blue!50}{$1$}]
  635 + (Ab) at (4,4) {$\co{a}b$};
  636 + % --------------------------------
  637 + \node [ sup,
  638 + pin=135:\textcolor{blue!50}{$1$}]
  639 + (Ac) at (5,4) {$\co{a}c$};
  640 + % --------------------------------
  641 + \node [ sup,
  642 + pin=135:\textcolor{blue!50}{$1$}]
  643 + (AB) at (6,4) {$\co{a}\co{b}$};
  644 + % --------------------------------
  645 + \node [ sup,
  646 + pin=135:\textcolor{blue!50}{$1$}]
  647 + (AC) at (7,4) {$\co{a}\co{c}$};
  648 + % --------------------------------
  649 + % \node [ ind,
  650 + % pin=270:\textcolor{purple}{$0$}]
  651 + % (bc) at (10,4) {$bc$};
  652 + % % --------------------------------
  653 + % \node [ ind,
  654 + % pin=270:\textcolor{purple}{$0$}]
  655 + % (bC) at (11,4) {$b\co{c}$};
  656 + % % --------------------------------
  657 + % \node [ ind,
  658 + % pin=270:\textcolor{purple}{$0$}]
  659 + % (Bc) at (9.5,3.5) {$\co{b}c$};
  660 + % % --------------------------------
  661 + % \node [ ind,
  662 + % pin=270:\textcolor{purple}{$0$}]
  663 + % (BC) at (10.5,3.5) {$\co{b}\co{c}$};
  664 + % --------------------------------
  665 + %
  666 + % --------------------------------
  667 + % \node [ sup,
  668 + % pin=90:\textcolor{blue!75}{$x\co{x}$}]
  669 + % (abc) at (1.5,6)
  670 + % {$abc$};
  671 + % % --------------------------------
  672 + % \node [ sup,
  673 + % pin=90:\textcolor{blue!50}{$x$}]
  674 + % (abC) at (0,6) {$ab\co{c}$};
  675 + % % --------------------------------
  676 + % \node [ sup,
  677 + % pin=90:\textcolor{blue!50}{$\co{x}$}]
  678 + % (aBc) at (3,6) {$a\co{b}c$};
  679 + % % --------------------------------
  680 + % \node [ ind,
  681 + % pin=90:\textcolor{purple}{$0$}]
  682 + % (aBC) at (5,6) {$a\co{b}\co{c}$};
  683 + % --------------------------------
  684 + \node [ sup,
  685 + pin=45:\textcolor{blue!50}{$1$}]
  686 + (Abc) at (7,6) {$\co{a}bc$};
  687 + % --------------------------------
  688 + \node [ sup,
  689 + pin=45:\textcolor{blue!50}{$1$}]
  690 + (AbC) at (8,6) {$\co{a}b\co{c}$};
  691 + % --------------------------------
  692 + \node [ sup,
  693 + pin=45:\textcolor{blue!50}{$1$}]
  694 + (ABc) at (9,6) {$\co{a}\co{b}c$};
  695 + % --------------------------------
  696 + \node [ sup,
  697 + pin=45:\textcolor{blue!50}{$1$}]
  698 + (ABC) at (10,6) {$\co{a}\co{b}\co{c}$};
  699 + % --------------------------------
  700 + %
  701 + % --------------------------------
  702 + % \draw [->] (ab) to [out=270,in=180] (E);
  703 + % \draw [->] (ab) to [out=270,in=90] (a);
  704 + % \draw [->] (ab) to [out=270,in=90] (b);
  705 + % \draw [->] (ab) to [out=90,in=270] (abc);
  706 + % \draw [->] (ab) to [out=90,in=270] (abC);
  707 + % %
  708 + % \draw [->] (ac) to [out=270,in=180] (E);
  709 + % \draw [->] (ac) to [out=270,in=90] (a);
  710 + % \draw [->] (ac) to [out=270,in=90] (c);
  711 + % \draw [->] (ac) to [out=90,in=270] (abc);
  712 + % \draw [->] (ac) to [out=90,in=270] (aBc);
384 713 %
385 714 \draw [->] (A) to [out=270,in=0] (E);
386 715 %
... ... @@ -415,22 +744,22 @@ rrect/.style={
415 744 w_d\at{x} = 0.\label{def:w.inconsistent}
416 745 \end{equation}
417 746 %
418   - \item $z$ is an interpretation and $\lset{z} = \set{z} = \uset{z}$. Then $z$ is a stable model and \textbf{define}
  747 + \item $z$ is an event and $\lset{z} = \set{z} = \uset{z}$. Then $z$ is a stable model and \textbf{define}
419 748 \begin{equation}
420 749 w_d\at{z} = w\at{z} = d\at{z} \pr{\theta_z}.\label{eq:prob.sm}
421 750 \end{equation}
422 751 %
423   - \item $z$ is an interpretation and $\lset{z} \neq \emptyset \land \uset{x} = \emptyset$. Then \textbf{define}
  752 + \item $z$ is an event and $\lset{z} \neq \emptyset \land \uset{x} = \emptyset$. Then \textbf{define}
424 753 \begin{equation}
425 754 w_d\at{z} = \sum_{s \in \lset{z}} w_d\at{s}.\label{def:w.disj}
426 755 \end{equation}
427 756 %
428   - \item $z$ is an interpretation and $\lset{z} = \emptyset \land \uset{z} \neq \emptyset$. Then \textbf{define}
  757 + \item $z$ is an event and $\lset{z} = \emptyset \land \uset{z} \neq \emptyset$. Then \textbf{define}
429 758 \begin{equation}
430 759 w_d\at{z} = \prod_{s \in \uset{z}} w_d\at{s}.\label{def:w.conj}
431 760 \end{equation}
432 761 %
433   - \item $z$ is an interpretation and $\lset{z} = \emptyset \land \uset{z} = \emptyset$. Then \textbf{define}
  762 + \item $z$ is an event and $\lset{z} = \emptyset \land \uset{z} = \emptyset$. Then \textbf{define}
434 763 \begin{equation}
435 764 w_d\at{z} = 0.\label{def:w.empty}
436 765 \end{equation}
... ... @@ -442,7 +771,7 @@ rrect/.style={
442 771 %
443 772 \item The last point defines a ``weight'' function on the samples that depends not only on the total choices and stable models of a PASP but also on a certain function $d$ that must respect some conditions. To simplify the notation we use the subscript in $w_d$ only when necessary.
444 773 %
445   - \item At first, it may seem counter-intuitive that $w\at{\emptyset} = \sum_{s\in\fml{S}} w\at{s}$ is the largest ``weight'' in the lattice. But $\emptyset$, as an interpretation, sets zero restrictions on the ``compatible'' stable models. The ``complement'' of $\bot = \emptyset$ is the \emph{maximal inconsistent} sample $\top = \fml{A} \cup \cset{\neg a }{ a \in \fml{A}}$.
  774 + \item At first, it may seem counter-intuitive that $w\at{\emptyset} = \sum_{s\in\fml{S}} w\at{s}$ is the largest ``weight'' in the lattice. But $\emptyset$, as an event, sets zero restrictions on the ``compatible'' stable models. The ``complement'' of $\bot = \emptyset$ is the \emph{maximal inconsistent} sample $\top = \fml{A} \cup \cset{\neg a }{ a \in \fml{A}}$.
446 775 %
447 776 \item \textbf{We haven't yet defined a probability measure.} To do so we must define a set of samples $\Omega$, a set of events $F\subseteq \pset{\Omega}$ and a function $P:F\to\intcc{0,1}$ such that:
448 777 \begin{enumerate}
... ... @@ -511,12 +840,12 @@ rrect/.style={
511 840 \end{frame}
512 841 % ================================================================
513 842 \begin{frame}
514   - Suppose that we add an annotation $\alpha :: a$, which entails $\co{\alpha} :: \neg a$. This is enough to get $w\at{s_1} = \co{\alpha}$ but, on the absence of further information, no fixed probability can be assigned to either model $s_2, s_3$ except that the respective sum must be $\alpha$. So, expressing our lack of knowledge using a parameter $d \in \intcc{0, 1}$ we get:
  843 + Suppose that we add an annotation $x :: a$, which entails $\co{x} :: \neg a$. This is enough to get $w\at{s_1} = \co{x}$ but, on the absence of further information, no fixed probability can be assigned to either model $s_2, s_3$ except that the respective sum must be $x$. So, expressing our lack of knowledge using a parameter $d \in \intcc{0, 1}$ we get:
515 844 $$
516 845 \begin{cases}
517   - w\at{s_1 } = &\co{\alpha}\cr
518   - w\at{s_2 } = &d\alpha\cr
519   - w\at{s_3} = &\co{d}\alpha.
  846 + w\at{s_1 } = &\co{x}\cr
  847 + w\at{s_2 } = &dx\cr
  848 + w\at{s_3} = &\co{d}x.
520 849 \end{cases}
521 850 $$
522 851 \end{frame}
... ... @@ -527,19 +856,19 @@ rrect/.style={
527 856 \begin{itemize}
528 857 \item Negations are represented as \emph{e.g.} $\co{a}$ instead of $\neg a$; Stable models are denoted by shaded nodes as \tikz{\node[fill=gray!50] {$ab$}}.
529 858  
530   - \item Interpretations in $\lset{x}$ are \emph{e.g.} \tikz{\node[draw, circle] {$a$}} and those in $\uset{x}$ are \emph{e.g.} \tikz{\node[draw] {$\co{a}b$}}. The remaining are simply denoted by \emph{e.g.} \tikz{\node {$a\co{b}$}}.
  859 + \item Events in $\lset{x}$ are \emph{e.g.} \tikz{\node[ circle] {$a$}} and those in $\uset{x}$ are \emph{e.g.} \tikz{\node[draw] {$\co{a}b$}}. The remaining are simply denoted by \emph{e.g.} \tikz{\node {$a\co{b}$}}.
531 860  
532   - \item The edges connect stable models with related interpretations. Up arrow indicate links to $\uset{s}$ and down arrows to $\lset{s}$.
  861 + \item The edges connect stable models with related events. Up arrow indicate links to $\uset{s}$ and down arrows to $\lset{s}$.
533 862  
534 863 \item The \emph{weight propagation} sets:
535 864 $$
536 865 \begin{aligned}
537   - w\at{abc} &= w\at{ab} w\at{ac} = \alpha^2d\co{d}, \cr
538   - w\at{\co{a}\cdot\cdot} &= w\at{\neg a} = \co{\alpha}, \cr
539   - w\at{a} &= w\at{ab} + w\at{ac} = \alpha(d + \co{d}) = \alpha, \cr
540   - w\at{b} &= w\at{ab} = d\alpha, \cr
541   - w\at{c} &= w\at{ac} = \co{d}\alpha, \cr
542   - w\at{\emptyset} &= w\at{ab} + w\at{ac} + w\at{\neg a} = d\alpha + \co{d}\alpha + \co{\alpha} = 1, \cr
  866 + w\at{abc} &= w\at{ab} w\at{ac} = x^2d\co{d}, \cr
  867 + w\at{\co{a}\cdot\cdot} &= w\at{\neg a} = \co{x}, \cr
  868 + w\at{a} &= w\at{ab} + w\at{ac} = x(d + \co{d}) = x, \cr
  869 + w\at{b} &= w\at{ab} = dx, \cr
  870 + w\at{c} &= w\at{ac} = \co{d}x, \cr
  871 + w\at{\emptyset} &= w\at{ab} + w\at{ac} + w\at{\neg a} = dx + \co{d}x + \co{x} = 1, \cr
543 872 w\at{a\co{b}} &= 0.
544 873 \end{aligned}
545 874 $$
... ... @@ -551,16 +880,16 @@ rrect/.style={
551 880 &+ w\at{a}+ w\at{b}+ w\at{c}\cr
552 881 &+ w\at{\emptyset}\cr
553 882 %
554   - &= - \alpha^{2} d^{2} + \alpha^{2} d + 2 \alpha d - 7 \alpha + 10
  883 + &= - x^{2} d^{2} + x^{2} d + 2 x d - 7 x + 10
555 884 \end{aligned}
556 885 $$
557   - \item Now, if $\alpha$ has an annotation to \emph{e.g.} $0.3$ we get
  886 + \item Now, if $x$ has an annotation to \emph{e.g.} $0.3$ we get
558 887 $$
559 888 Z = - 0.09 d^{2} + 0.69 d + 7.9
560 889 $$
561 890 \item Now some statistics are possible. For example we get
562 891 $$
563   - \pr{abc \mid \alpha = 0.3} = \frac{0.09 d \left(d - 1\right)}{0.09 d^{2} - 0.69 d - 7.9}
  892 + \pr{abc \mid x = 0.3} = \frac{0.09 d \left(d - 1\right)}{0.09 d^{2} - 0.69 d - 7.9}
564 893 $$.
565 894  
566 895 \item This expression can be plotted for $d\in\intcc{0,1}$
... ... @@ -571,7 +900,7 @@ rrect/.style={
571 900 \item If a data set $E$ entails \emph{e.g.} $\pr{abc \mid E} = 0.0015$ we can numerically solve
572 901 $$
573 902 \begin{aligned}
574   - \pr{abc \mid \alpha = 0.3} &= \pr{abc \mid E} \cr
  903 + \pr{abc \mid x = 0.3} &= \pr{abc \mid E} \cr
575 904 \iff\cr
576 905 \frac{0.09 d \del{d - 1}}{0.09 d^{2} - 0.69 d - 7.9} &= 0.0015
577 906 \end{aligned}
... ... @@ -613,9 +942,9 @@ rrect/.style={
613 942 % ================================================================
614 943 \begin{frame}
615 944  
616   - Looking into probabilistic interpretations of the program and/or its models, we define $\alpha = \pr{\Theta = \theta_1}\in\intcc{0, 1}$ and $\pr{\Theta = \theta_2} = \co{\alpha}$.
  945 + Looking into probabilistic events of the program and/or its models, we define $x = \pr{\Theta = \theta_1}\in\intcc{0, 1}$ and $\pr{\Theta = \theta_2} = \co{x}$.
617 946  
618   - Since $s_1$ is the only stable model that results from $\Theta = \theta_1$, it is natural to extend $\pr{ s_1 } = \pr{\Theta = \theta_1} = \alpha$. However, there is no clear way to assign $\pr{s_2}, \pr{s_3}$ since \emph{both models result from the single total choice} $\Theta = \theta_2$. Clearly,
  947 + Since $s_1$ is the only stable model that results from $\Theta = \theta_1$, it is natural to extend $\pr{ s_1 } = \pr{\Theta = \theta_1} = x$. However, there is no clear way to assign $\pr{s_2}, \pr{s_3}$ since \emph{both models result from the single total choice} $\Theta = \theta_2$. Clearly,
619 948 $$\pr{s_2 \mid \Theta} + \pr{s_3 \mid \Theta} =
620 949 \begin{cases}
621 950 0 & \text{if}~\Theta = \theta_1\cr
... ... @@ -639,13 +968,13 @@ rrect/.style={
639 968 \begin{array}{cc|l}
640 969 A, B, C& P & \text{Obs.}\cr
641 970 \hline
642   - a, \neg b, c & \alpha & s_1, \Theta=\theta_1\cr
643   - \neg a, b, \neg c & \co{\alpha}\beta & s_2, \Theta=\theta_2\cr
644   - \neg a, \neg b, c & \co{\alpha}\co{\beta} & s_3, \Theta=\theta_2\cr
  971 + a, \neg b, c & x & s_1, \Theta=\theta_1\cr
  972 + \neg a, b, \neg c & \co{x}\beta & s_2, \Theta=\theta_2\cr
  973 + \neg a, \neg b, c & \co{x}\co{\beta} & s_3, \Theta=\theta_2\cr
645 974 \ast & 0&\text{not stable models}
646 975 \end{array}
647 976 $$
648   - where $\alpha, \beta\in\intcc{0,1}$.
  977 + where $x, \beta\in\intcc{0,1}$.
649 978 \end{frame}
650 979 % ================================================================
651 980 \section{Conclusions}
... ... @@ -680,14 +1009,14 @@ rrect/.style={
680 1009  
681 1010 \item The \deft{Herbrand base} of a program is the set of ground literals that result from combining all the predicates and constants of the program.
682 1011 \begin{itemize}
683   - \item An \deft{interpretation} is a consistent subset (\emph{i.e.} doesn't contain $\set{a, \neg a}$) of the Herbrand base.
684   - \item Given an interpretation $I$, a ground literal $a$ is \deft{true}, $I \models a$, if $a \in I$; otherwise the literal is \deft{false}.
  1012 + \item An \deft{event} is a consistent subset (\emph{i.e.} doesn't contain $\set{a, \neg a}$) of the Herbrand base.
  1013 + \item Given an event $I$, a ground literal $a$ is \deft{true}, $I \models a$, if $a \in I$; otherwise the literal is \deft{false}.
685 1014 \item A ground subgoal, $\naf b$, where $b$ is a ground literal, is \deft{true}, $I \models \naf b$, if $b \not\in I$; otherwise, if $b \in I$, it is \deft{false}.
686   - \item A ground rule $r = h_1 \vee \cdots \vee h_m \leftarrow b_1 \wedge \cdots \wedge b_n$ is \deft{satisfied} by the interpretation $I$, \emph{i.e.} $I \models r$, iff
  1015 + \item A ground rule $r = h_1 \vee \cdots \vee h_m \leftarrow b_1 \wedge \cdots \wedge b_n$ is \deft{satisfied} by the event $I$, \emph{i.e.} $I \models r$, iff
687 1016 $$
688 1017 \forall j \exists i~I \models b_j \implies I \models h_i.
689 1018 $$
690   - \item A \deft{model} of a program is an interpretation that satisfies all its rules. Denote $\fml{M}_P$ the set of all models of $P$.
  1019 + \item A \deft{model} of a program is an event that satisfies all its rules. Denote $\fml{M}_P$ the set of all models of $P$.
691 1020 \end{itemize}
692 1021  
693 1022 \item The \deft{dependency graph} of a program is a digraph where:
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701 1030 \end{itemize}
702 1031 \item Every \emph{definite program} has a unique minimal model: its \deft{semantic}.
703 1032 \item Programs with negation may have no unique minimal model.
704   - \item Given a program $P$ and an interpretation $I$, their \deft{reduct}, $P^I$, is the propositional program that results from
  1033 + \item Given a program $P$ and an event $I$, their \deft{reduct}, $P^I$, is the propositional program that results from
705 1034 \begin{enumerate}
706 1035 \item Removing all the rules with $\naf b$ in the body where $b \in I$.
707 1036 \item Removing all the $\naf b$ subgoals from the remaining rules.
708 1037 \end{enumerate}
709   - \item A \deft{stable model} (or \deft{answer set}) of the program $P$ is an interpretation $I$ that is the minimal model of the reduct $P^I$.
  1038 + \item A \deft{stable model} (or \deft{answer set}) of the program $P$ is an event $I$ that is the minimal model of the reduct $P^I$.
710 1039 \item Denote $\fml{S}_P$ the set of all stable models of program $P$. The \deft{semantics} (or \deft{answer sets}) of a program $P$ is the set $\fml{S}_P$.
711 1040 \begin{itemize}
712 1041 \item Some programs, such as $a \leftarrow \naf a$, have no stable models.
713   - \item A stable model is an interpretation closed under the rules of the program.
  1042 + \item A stable model is an event closed under the rules of the program.
714 1043 \end{itemize}
715 1044 \end{itemize}
716 1045 \end{frame}
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1   -\beamer@sectionintoc {1}{Introduction}{2}{0}{1}
2   -\beamer@sectionintoc {2}{Extending Probability to Samples}{11}{0}{2}
3   -\beamer@sectionintoc {3}{Cases \& Examples}{18}{0}{3}
4   -\beamer@subsectionintoc {3}{1}{Programs with disjunctive heads}{19}{0}{3}
5   -\beamer@subsectionintoc {3}{2}{Non-stratified programs}{23}{0}{3}
6   -\beamer@sectionintoc {4}{Conclusions}{28}{0}{4}
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  1 +\documentclass[a4paper]{article}
  2 +\usepackage{hyperref}
  3 +\hypersetup{
  4 + colorlinks=true,
  5 + linkcolor=blue,
  6 +}
  7 +\usepackage{commath}
  8 +\usepackage{amssymb}
  9 +%
  10 +% Local commands
  11 +%
  12 +\newcommand{\todo}[1]{{\color{orange}TODO #1}}
  13 +\newcommand{\naf}{\ensuremath{\sim\!}}
  14 +\newcommand{\larr}{\ensuremath{\leftarrow}}
  15 +\newcommand{\at}[1]{\ensuremath{\!\del{#1}}}
  16 +\newcommand{\co}[1]{\ensuremath{\overline{#1}}}
  17 +\newcommand{\fml}[1]{\ensuremath{{\cal #1}}}
  18 +\newcommand{\deft}[1]{\textbf{#1}}
  19 +\newcommand{\pset}[1]{\ensuremath{\mathbb{P}\at{#1}}}
  20 +\newcommand{\ent}{\ensuremath{\lhd}}
  21 +\newcommand{\cset}[2]{\ensuremath{\set{#1,~#2}}}
  22 +\newcommand{\langof}[1]{\ensuremath{\fml{L}\at{#1}}}
  23 +\newcommand{\uset}[1]{\ensuremath{\left|{#1}\right>}}
  24 +\newcommand{\lset}[1]{\ensuremath{\left<{#1}\right|}}
  25 +\newcommand{\pr}[1]{\ensuremath{\mathrm{p}\at{#1}}}
  26 +\newcommand{\given}{\ensuremath{~\middle|~}}
  27 +
  28 +\title{Zugzwang\\\textit{Logic and Artificial Intelligence}}
  29 +\author{Francisco Coelho\\ \texttt{fc@uevora.pt}}
  30 +
  31 +\begin{document}
  32 +\maketitle
  33 +
  34 +\begin{abstract}
  35 + A major limitation of logical representations is the implicit assumption that the Background Knowledge (BK) is perfect. This assumption is problematic if data is noisy, which is often the case. Here we aim to explore how ASP specifications with probabilistic facts can lead to characterizations of probability functions on the specification's domain.
  36 +\end{abstract}
  37 +
  38 +\section{Introduction and Motivation }
  39 +
  40 +Answer Set Programming (ASP) is a logic programming paradigm based on the Stable Model semantics of Normal Logic Programs (NP) that can be implemented using the latest advances in SAT solving technology. ASP is a truly declarative language that supports language constructs such as disjunction in the head of a clause, choice rules, and hard and weak constraints.
  41 +
  42 +The Distribution Semantics (DS) is a key approach to extend logical representations with probabilistic reasoning. Probabilistic Facts (PF) are the most basic stochastic DS primitive and they take the form of logical facts, $a$, labelled with a probability, such as $p::a$; Each probabilistic fact represents a boolean random variable that is true with probability $p$ and false with probability $1 - p$. A (consistent) combination of the PFs defines a \textit{total choice} $c = \set{p::a, \ldots}$ such that
  43 +
  44 +\begin{equation}
  45 + P(c) = \prod_{a\in c} p \prod_{a \not\in c} (1- p).
  46 +\end{equation}
  47 +
  48 +Our goal is to extend this probability, from total choices, to cover the specification domain. We can foresee two key applications of this extended probability:
  49 +
  50 +\begin{enumerate}
  51 + \item Support any probabilistic reasoning/task on the specification domain.
  52 + \item Also, given a dataset and a divergence measure, now the specification can be scored (by the divergence w.r.t. the \emph{empiric} distribution of the dataset), and sorted amongst other specifications. This is a key ingredient in algorithms searching, for example, an \textit{optimal specification} of the dataset.
  53 +\end{enumerate}
  54 +
  55 +This goal faces a critical problem concerning situations where \textit{multiple} standard models result from a given total choice, illustrated by the following example. The specification
  56 +$$
  57 +\begin{aligned}
  58 + 0.3::a&,\cr
  59 + b \vee c& \leftarrow a.
  60 +\end{aligned}
  61 +$$
  62 +has three stable models, $\set{\neg a}, \set{a, b}$ and $\set{a, c}$. While it is straightforward to set $P(\neg a)=0.7$, there is \textit{no further information} to assign values to $P(a,b)$ and $P(a,c)$. At best, we can use a parameter $\alpha$ such that
  63 +$$
  64 +\begin{aligned}
  65 +P(a,b) &= 0.3 \alpha,\cr
  66 +P(a,c) &= 0.3 (1 - \alpha).
  67 +\end{aligned}
  68 +$$
  69 +
  70 +This uncertainty in inherent to the specification, but can be mitigated with the help of a dataset: the parameter $\alpha$ can be estimated from the empirical distribution.
  71 +
  72 +In summary, if an ASP specification is intended to describe some system that can be observed then:
  73 +
  74 +\begin{enumerate}
  75 + \item The observations can be used to estimate the value of the parameters (such as $\alpha$ above and others entailed from further clauses).
  76 + \item With a probability set for the stable models, we want to extend it to all the samples (\textit{i.e.} consistent sets of literals) of the specification.
  77 + \item This extended probability can then be related to the \textit{empirical distribution}, using a probability divergence, such as Kullback-Leibler; and the divergence value used as a \textit{performance} measure of the specification with respect to the observations.
  78 + \item If that specification is only but one of many possible candidates then that performance measure can be used, \textit{e.g.} as fitness, by algorithms searching (optimal) specifications of a dataset of observations.
  79 +\end{enumerate}
  80 +
  81 +Currently, we are on the step two above: Extending a probability function (with parameters such as $\alpha$), defined on the stable sets of a specification, to all the samples of the specification. This extension must, of course, respect the axioms of probability so that probabilistic reasoning is consistent with the ASP specification.
  82 +
  83 +\section{Extending Probabilities}
  84 +
  85 +Given an ASP specification, we look at the \textit{total choices} $\fml{C}$, \textit{stable models} $\fml{S}$, \textit{atoms} and \textit{literals} $\fml{A}, \fml{L}$, \textit{samples} $z \in \fml{Z} \iff z \subseteq \fml{L}$ and \textit{events} $\fml{E}$ (consistent samples).
  86 +
  87 +\begin{enumerate}
  88 + \item \textbf{Given} $\pr{C = c}$ from the specification. Each total choice $C = c$ (together with the specification facts and rules) entails some stable models, $S_c$.
  89 + \item \textbf{Given} $\pr{S = s \given C = c}$ such that its value is zero if $s \not\in S_c$.
  90 + \item Each event $E = e$ either:
  91 + \begin{enumerate}
  92 + \item Is a stable model. Then
  93 + \begin{equation}
  94 + \pr{E = e \given C = c} = \pr{S = e \given C = c}.
  95 + \end{equation}
  96 + \item \textit{Is contained} in some stable models. Then
  97 + \begin{equation}
  98 + \pr{E = e \given C = c} = \sum_{s \supseteq e}\pr{S = s \given C = c}.
  99 + \end{equation}
  100 + \item \textit{Contains} some stable models. Then
  101 + \begin{equation}
  102 + \pr{E = e \given C = c} = \prod_{s \subseteq e}\pr{S = s \given C = c}.
  103 + \end{equation}
  104 + \item Neither contains nor is contained by a stable model. Then
  105 + \begin{equation}
  106 + \pr{E = e \given C = c} = 0.
  107 + \end{equation}
  108 + \end{enumerate}
  109 + \item For each sample $Z = z$
  110 + \begin{equation}
  111 + \pr{Z = z \given C = c} = \begin{cases}
  112 + \pr{E = z \given C = c} & z \in \fml{E}, \cr
  113 + 0 & \text{otherwise}.
  114 + \end{cases}
  115 + \end{equation}
  116 +\end{enumerate}
  117 +
  118 +\begin{quotation}\marginpar{Todo} Prove the four event cases, and support the sum/product options, with the independence assumptions.
  119 +\end{quotation}
  120 +
  121 +
  122 +
  123 +\section*{References}
  124 +
  125 +
  126 +\begin{enumerate}
  127 + \item Victor Verreet, Vincent Derkinderen, Pedro Zuidberg Dos Martires, Luc De Raedt, Inference and Learning with Model Uncertainty in Probabilistic Logic Programs (2022)
  128 + \item Andrew Cropper, Sebastijan Dumancic, Richard Evans, Stephen H. Muggleton, Inductive logic programming at 30 (2021)
  129 + \item Fabio Gagliardi Cozman, Denis Deratani Mauá, The joy of Probabilistic Answer Set Programming: Semantics - complexity, expressivity, inference (2020)
  130 + \item Fabrizio Riguzzi, Foundations of Probabilistic Logic Programming Languages, Semantics, Inference and Learning. Rivers Publishers (2018)
  131 + \item Martin Gebser, Roland Kaminski, Benjamin Kaufmann, and Torsten Schaub, Answer Set Solving in Practice, Morgan \& Claypool Publishers (2013)
  132 +\end{enumerate}
  133 +
  134 +\end{document}
0 135 \ No newline at end of file
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